Equivariant hyperbolization of 3-manifolds via homology cobordisms
Equivariant hyperbolization of 3-manifolds via homology cobordisms
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DOI：
10.1016/j.topol.2023.108485
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发表时间：
2018-04
影响因子：
0.6
通讯作者：
D. Auckly;H. Kim;P. Melvin;Daniel Ruberman
中科院分区：
文献类型：
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作者：
D. Auckly;H. Kim;P. Melvin;Daniel Ruberman
本文的主要结果是：任何具有有限群作用的三维流形都是双曲流形的等价可逆同调余界；当扭曲系数适当时，这一结果也成立。以下两个结果推动了这项工作。首先，对于一大类有限群，存在双曲等变软木(如作者以前的工作中所定义的)。其次，作用在同调3-球面上的任何有限群也作用在双曲同调3-球面上。这个定理还有其他推论，包括存在无限多个支持不在任何可压缩流形上扩张的自由Z p-作用的双曲同调球，以及(从该定理的非等变版本)无限多个限制同调球但不限制可压缩流形的双曲同调球面。同时，证明了三维流形上的可逆同调余边关系是反对称的，从而是一个偏序关系。
The main result of this paper is that any 3-dimensional manifold with a finite group action is equivariantly invertibly homology cobordant to a hyperbolic manifold; this result holds with suitable twisted coefficients as well. The following two consequences motivated this work. First, there are hyperbolic equivariant corks (as defined in previous work of the authors) for a wide class of finite groups. Second, any finite group that acts on a homology 3-sphere also acts on a hyperbolic homology 3-sphere. The theorem has other corollaries, including the existence of infinitely many hyperbolic homology spheres that support free Z p-actions that do not extend over any contractible manifolds, and (from the non-equivariant version of the theorem) infinitely many that bound homology balls but do not bound contractible manifolds. In passing, it is shown that the invertible homology cobordism relation on 3-manifolds is antisymmetric, and thus a partial order.